Solve Using the Quadratic Formula x^4-4x^2+2=0

$x^{4} - 4 x^{2} + 2 = 0$

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 4 u + 2 = 0$

$u = x^{2}$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 4$ , and $c = 2$ into the quadratic formula and solve for $u$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$u = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$u = \frac{4 \pm \sqrt{16 - 4 \cdot 2}}{2 \cdot 1}$

Multiply $- 4$ by $2$ .

$u = \frac{4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Subtract $8$ from $16$ .

$u = \frac{4 \pm \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$u = \frac{4 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$u = \frac{4 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$u = \frac{4 \pm 2 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$u = \frac{4 \pm 2 \sqrt{2}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{2}}{2}$ .

$u = 2 \pm \sqrt{2}$

The final answer is the combination of both solutions.

$u = 2 + \sqrt{2}, 2 - \sqrt{2}$

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 3.41421356$

${(x^{2})}^{1} = 0.58578643$

Solve the first equation for $x$ .

$x^{2} = 3.41421356$

Solve the equation for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{3.41421356}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{3.41421356}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{3.41421356}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{3.41421356}, - \sqrt{3.41421356}$

Solve the second equation for $x$ .

${(x^{2})}^{1} = 0.58578643$

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = 0.58578643$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0.58578643}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{0.58578643}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{0.58578643}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{0.58578643}, - \sqrt{0.58578643}$

The solution to $x^{4} - 4 x^{2} + 2 = 0$ is $x = \sqrt{3.41421356}, - \sqrt{3.41421356}, \sqrt{0.58578643}, - \sqrt{0.58578643}$ .

$x = \sqrt{3.41421356}, - \sqrt{3.41421356}, \sqrt{0.58578643}, - \sqrt{0.58578643}$

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3.41421356}, - \sqrt{3.41421356}, \sqrt{0.58578643}, - \sqrt{0.58578643}$

Decimal Form:

$x = 1.84775906 \dots, - 1.84775906 \dots, 0.76536686 \dots, - 0.76536686 \dots$

Solve Using the Quadratic Formula x^4-4x^2+2=0

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